On an extensible beam equation with nonlinear damping and source terms

The paper studies the global existence, stability and the longtime dynamics of solutions to the initial boundary value problem (IBVP) of an extensible beam equation with nonlinear damping and source terms: utt−M(‖∇u‖2)Δu+Δ2u+g(ut)+f(u)=h. It proves that (i) the IBVP is global well posed provided that either the growth exponent p of the source term f(u) is non-supercritical, that is, and p⩽q if N⩾5 or p is supercritical but is dominated by the growth exponent q of the nonlinear damping g(ut), i.e. p⁎p⁎ if N⩾5; (iii) especially, when the space dimension N⩽4, all the above mentioned conclusions hold without any restriction on p and q, that is, 1⩽p,q<∞.